Stability and robust stabilisation for a class of non-linear uncertain discrete-time descriptor Markov jump systems

The stability and robust state feedback stabilisation problems for a class of non-linear discrete-time descriptor Markov jump systems with parameter uncertainties are investigated. The non-linear function satisfies a quadratic constraint. First, based on Lyapunov theory and the implicit function theorem, a linear matrix inequality (LMI) sufficient condition is developed, which guarantees that the non-linear discrete-time descriptor Markov jump systems are regular, causal, and have unique solution in a neighbourhood of the origin, and are stochastically stable. With this condition, another LMI condition is developed based on the singular value decomposition approach, which guarantees that the systems are regular, causal, have unique solution in a neighbourhood of the origin, and are stochastically stable. Then, the design method of robust state feedback controllers is given. Last, numerical examples are provided to illustrate the effectiveness of the proposed methods.